With regard to the case in "Research Corner" concerning number which can be written as the sum of three squares and four squares, all numbers of the form $m^{2n}(8k+6)$ can only be written as the sum of at least three square numbers
With regard to the case in "Research Corner" concerning number which can be written as the sum of three squares and four squares, all numbers of the form $m^{2n}(8k+6)$ can only be written as the sum of at least three square numbers